The numerical solution of nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations via the meshless method of integrated radial basis functions

Ebrahimijahan, Ali; Dehghan, Mehdi

doi:10.1007/s00366-019-00811-3

Cited by 16 publications

(4 citation statements)

References 49 publications

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“…Dehghan and Shafieeabyaneh addressed the RLW and extended Fisher-Kolmogorov (EFK) equations using local meshless RBF-FD [59]. Ebrahimijahan and Dehghan proposed a numerical technique for solving the nonlinear generalized BBBM-B and RLW equations based on the integrated RBF [60]. Rashidinia et al implemented the local RBF-FD meshless method for generalized Korteweg-de Vries-Burgers [61] and Kawahara [62] equations.…”

Section: The Rbf-fd Collocation Methodsmentioning

confidence: 99%

Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation

2020

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This paper investigates the solitary wave solutions of the generalized Rosenau–Korteweg-de Vries-regularized-long wave equation. This model is obtained by coupling the Rosenau–Korteweg-de Vries and Rosenau-regularized-long wave equations. The solution of the equation is approximated by a local meshless technique called radial basis function (RBF) and the finite-difference (FD) method. The association of the two techniques leads to a meshless algorithm that does not requires the linearization of the nonlinear terms. First, the partial differential equation is transformed into a system of ordinary differential equations (ODEs) using radial kernels. Then, the ODE system is solved by means of an ODE solver of higher-order. It is shown that the proposed method is stable. In order to illustrate the validity and the efficiency of the technique, five problems are tested and the results compared with those provided by other schemes.

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Section: The Rbf-fd Collocation Methodsmentioning

confidence: 99%

Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation

2020

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“…The RBF approximations can be constructed through differentiation (DRBF) or integration (IRBF). The governing equations of fluid dynamics have been successfully solved by the DRBF- and IRBF-based methods (Mai-Duy and Tanner, 2005, 2007; Dehghan and Shokri, 2008; Kosec and Šarler, 2008, 2013; Mohebbi et al , 2014; Ngo-Cong et al , 2017; Ebrahimijahan and Dehghan, 2021; Ebrahimijahan et al , 2020, 2022a, 2022b; Abbaszadeh et al , 2022; Mesgarani et al , 2022).…”

Section: Introductionmentioning

confidence: 99%

An effective high-order five-point stencil, based on integrated-RBF approximations, for the first biharmonic equation and its applications in fluid dynamics

Mai‐Duy

Tien

Strunin

et al. 2023

HFF

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Purpose The purpose of this paper is to present a new discretisation scheme, based on equation-coupled approach and high-order five-point integrated radial basis function (IRBF) approximations, for solving the first biharmonic equation, and its applications in fluid dynamics. Design/methodology/approach The first biharmonic equation, which can be defined in a rectangular or non-rectangular domain, is replaced by two Poisson equations. The field variables are approximated on overlapping local regions of only five grid points, where the IRBF approximations are constructed to include nodal values of not only the field variables but also their second-order derivatives and higher-order ones along the grid lines. In computing the Dirichlet boundary condition for an intermediate variable, the integration constants are used to incorporate the boundary values of the first-order derivative into the boundary IRBF approximation. Findings These proposed IRBF approximations on the stencil and on the boundary enable the boundary values of the derivative to be exactly imposed, and the IRBF solution to be much more accurate and not influenced much by the RBF width. The error is reduced at a rate that is much greater than four. In fluid dynamics applications, the method is able to capture well the structure of steady highly non-linear fluid flows using relatively coarse grids. Originality/value The main contribution of this study lies in the development of an effective high-order five-point stencil based on IRBFs for solving the first biharmonic equation in a coupled set of two Poisson equations. A fast rate of convergence (up to 11) is achieved.

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“…Bhardwaj and Kumar [15] used the FDM-based mesh less approach for the numerical solution of the nonlinear diffusion-wave equation. Based on the integrated radial basis functions and a finite difference scheme, the regularized long-wave equations are investigated in [16,17]. Oruc [18] examined numerical solutions of Zakharov-Rubenchik system by using radial basis function finite difference (RBF-FD) mesh less method and an explicit Runge-Kutta method.…”

Section: Introductionmentioning

confidence: 99%

A Novel Meshfree Strategy for a Viscous Wave Equation With Variable Coefficients

et al. 2021

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A one-step new general mesh free scheme, which is based on radial basis functions, is presented for a viscous wave equation with variable coefficients. By constructing a simple extended radial basis function, it can be directly applied to wave propagation by using the strong form-based mesh free collocation method. There is no need to deal with the time-dependent variable particularly. Numerical results for a viscous wave equation with variable coefficients show that the proposed mesh free collocation method is simple with accurate solutions.

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The numerical solution of nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations via the meshless method of integrated radial basis functions

Cited by 16 publications

References 49 publications

Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation

Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation

An effective high-order five-point stencil, based on integrated-RBF approximations, for the first biharmonic equation and its applications in fluid dynamics

A Novel Meshfree Strategy for a Viscous Wave Equation With Variable Coefficients

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